Fractional Spin through Quantum Affine Algebra $\hat A(n)$ and quantum affine superalgebra $\hat A(n,m)$

TL;DR

This paper explores the fractional decomposition of quantum affine algebras and superalgebras using oscillator representations, revealing connections to fractional supersymmetry and establishing equivalence with classical algebras.

Contribution

It introduces a method to decompose quantum affine algebras and superalgebras into fractional parts using Q-deformed bosons and oscillator representations, linking to fractional supersymmetry.

Findings

Fractional decomposition of $\hat A(n)$ and $\hat A(n,m)$ achieved in the Q→q limit.

Establishment of the relation between fractional decomposition and fractional supersymmetry.

Proven equivalence between quantum affine algebra $\hat A(n)$ and classical algebra in fermionic realization.

Abstract

Using the splitting of a $Q$-deformed boson, in the $Q \to q= e^{\frac{\rm 2\pi i}{\rm k}}$ limit, the fractional decomposition of the quantum affine algebra $\hat A(n)$ and the quantum affine superalgebra $\hat A(n,m)$ are found. This decomposition is based on the oscillator representation and can be related to the fractional supersymmetry and k-fermionic spin. We establish also the equivalence between the quantum affine algebra $\hat A(n)$ and the classical one in the fermionic realization.